TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 357, Number 4, Pages 1627–1661 S 0002-9947(04)03575-5 Article electronically published on October 28, 2004 THE TITS BOUNDARY OF A CAT(0) 2-COMPLEX XIANGDONG XIE Abstract. We investigate the Tits boundary of CAT(0) 2-complexes that have only a finite number of isometry types of cells. In particular, we show that away from the endpoints, a geodesic segment in the Tits boundary is the ideal boundary of an isometrically embedded Euclidean sector. As applications, we provide sufficient conditions for two points in the Tits boundary to be the endpoints of a geodesic in the 2-complex and for a group generated by two hyperbolic isometries to contain a free group. We also show that if two CAT(0) 2-complexes are quasi-isometric, then the cores of their Tits boundaries are bi- Lipschitz. 1. Introduction In this paper we study the Tits boundary of CAT(0) 2-complexes that have only a finite number of isometry types of cells. A CAT(0) 2-complex in this paper is a CAT(0) piecewise Riemannian 2-complex. Here each closed 2-cell is equipped with a Riemannian metric so that it is convex and its boundary is a broken geodesic. The metric on the 2-complex is the induced path metric. Hadamard manifolds are simply connected complete Riemannian manifolds of nonpositive sectional curvature. CAT(0) spaces are counterparts of Hadamard manifolds in the category of metric spaces. A CAT(0) space is a complete sim- ply connected geodesic metric space so that all its triangles are at least as thin as the triangles in the Euclidean space. CAT(0) spaces have many of the geo- metric properties enjoyed by Hadamard manifolds including convexity of distance functions, uniqueness of geodesic segments and contractibility. As in the case of Hadamard manifolds, a CAT(0) space X has a well-defined ideal boundary ∂∞X. There is a topology on ∂∞X called the cone topology and a metric dT on ∂∞X called the Tits metric. The topology induced by dT is usually different from the cone topology. Given a CAT(0) space X, the Tits boundary ∂T X means the ideal boundary equipped with the Tits metric dT . ∂T X reflects the large scale geometry of X.In particular, ∂T X encodes information on large flat subspaces of X as well as the amount of negative curvature in X (hyperbolicity). Tits metric and Tits boundary are closely related to many interesting questions in geometry. They play an impor- tant role in many rigidity theorems (J. Heber [H], W. Ballmann [B], G. Mostow [M], B. Leeb [L]). They are also very important to the question of whether a group Received by the editors March 10, 2003 and, in revised form, December 1, 2003. 2000 Mathematics Subject Classification. Primary 20F67, 20F65; Secondary 57M20, 53C20. Key words and phrases. Tits boundary, Tits metric, CAT(0), 2-complex, quasi-isometry, quasi- flat. c 2004 American Mathematical Society 1627 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1628 XIANGDONG XIE of isometries of X contains a free group of rank two (K. Ruane [R], X. Xie [X1], [X2]). For an arbitrary CAT(0) space X, ∂T X is complete and CAT(1) (see Section 2.4 or [B]). The Tits boundary of the product of two CAT(0) spaces is the spherical join of the Tits boundaries of the two factors. When a locally compact CAT(0) space X admits a cocompact isometric action, B. Kleiner ([K]) showed the geometric di- mension of ∂T X is 1 less than the maximal dimension of isometrically embedded Euclidean spaces in X. Aside from these general results not much is known about the Tits boundary of CAT(0) spaces. Tits boundary is not even well understood for Hadamard manifolds or 2-dimensional (nonmanifold) CAT(0) spaces. Tits bound- ary is well understood only for a few classes of CAT(0) spaces: ∂T X is discrete if X is a Gromov hyperbolic CAT(0) space; ∂T X is a spherical building if X is a higher rank symmetric space or Euclidean building; ∂T X contains interval components if X is the universal cover of a nonpositively curved graph manifold (S. Buyalo and V. Schroeder [BS], C. Croke and B. Kleiner [CK2]); the Tits boundary was also studied for real analytic Hadamard 4-manifolds admitting cocompact actions (C. Hummel and V. Schroeder [HS1], [HS2]) and the universal covers of certain torus complexes (C. Croke and B. Kleiner [CK1]). Let X be a CAT(0) 2-complex. B. Kleiner’s theorem (Theorem 7.1 of [K]) implies that ∂T X has geometric dimension at most 1. It follows that any closed metric ball with radius r (r<π/2) in ∂T X is an R-tree. For an R-treewecan talk about geodesic segments and branch points. Our first goal is to understand geodesic segments and branch points in the Tits boundary. A sector is a closed convex subset of the Euclidean plane E2 whose boundary is the union of two rays emanating from the origin. We equip a sector with the induced metric. The image of an isometric embedding from a sector into a CAT(0) space X is called a flat sector in X.WenoticeifS is a flat sector in a CAT(0) space X, then the Tits boundary ∂T S of S is a closed interval and isometrically embeds into the Tits boundary ∂T X of X. The following theorem says that away from the endpoints, a segment in the Tits boundary is the Tits boundary of a flat sector. For a CAT(0) 2-complex X, we say Shape(X) is finite if X has only finitely many isometry types of cells. Theorem 3.1. Let X be a CAT(0) 2-complex with Shape(X) finite, and γ :[0,h] → ∂T X ageodesicin∂T X with length h ≤ π. Then for any >0, there exists a flat sector S in X with ∂T S = γ([, h − ]). Theorem 3.1 does not hold without the assumption that Shape(X) is finite (see the example in Section 3). Theorem 3.1 also cannot be improved to include the case =0:letX be the universal cover of a torus complex considered in [CK1], then ∂T X contains interval components; some of these intervals are not the Tits boundaries of any flat sectors. It follows from Theorem 3.1 that (see Proposition 4.12) a branch point in ∂T X is represented by a ray where two flat sectors branch off. Let X be a CAT(0) 2-complex with Shape(X) finite. Then small metric balls in ∂T X are R-trees. An R-tree may have lots of branch points. One may ask if there is a constant c = c(X) > 0 such that the distance between any two branch points in ∂T X is at least c. Another interesting question concerning the Tits boundary is whether the lengths of circles in ∂T X form a discrete set. Note B. Kleiner’s theorem also implies circles in ∂T X are simple closed geodesics, hence are rectifiable. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE TITS BOUNDARY OF A CAT(0) 2-COMPLEX 1629 Theorem 4.10 and Theorem 4.13. Let X be a locally finite CAT(0) 2-complex with Shape(X) finite. Suppose the interior angles of all the closed 2-cells of X are rational multiples of π. Then there is a positive integer m such that: (i) each topological circle in ∂T X has length an integral multiple of π/m; (ii) the distance between any two branch points in ∂T X is either infinite or an integral multiple of π/m. Given a geodesic c : R → X in a CAT(0) space X, the two points in the ideal boundary determined by c|[0,∞) and c|(−∞,0] are called the endpoints of c.As an application of Theorem 3.1 we discuss when two points ξ,η ∈ ∂∞X are the endpoints of a geodesic in a locally compact CAT(0) space X. Recall a necessary condition is dT (ξ,η) ≥ π, and a sufficient condition is dT (ξ,η) >π.Weprovidea criterion for ξ and η to be the endpoints of a geodesic in X when X is a CAT(0) 2-complex and dT (ξ,η)=π. A point in the ideal boundary of a CAT(0) space is called a terminal point if it does not lie in the interior of any Tits geodesic. Theorem 4.15. Let X be a locally finite CAT(0) 2-complex with Shape(X) finite. If ξ,η ∈ ∂∞X are not terminal points and dT (ξ,η) ≥ π, then there is a geodesic in X with ξ and η as endpoints. As a further application we provide a sufficient condition for a group generated by two hyperbolic isometries to contain a free group of rank two. Recall that an isometry g : X → X is a hyperbolic isometry if there is a geodesic c : R → X and a positive number l such that g(c(t)) = c(t + l) for all t.Thegeodesicc is called an axis of g. Denote the two endpoints of the axis c by g(+∞)andg(−∞). Theorem 5.3. Let X be a locally finite CAT(0) 2-complex with Shape(X) finite, and g1, g2 two cellular hyperbolic isometries of X. Suppose each closed 2-cell of X is isometric to a convex polygon in the Euclidean plane.